Let's just define your expression for future use:

expr[r_, t_, \[Theta]_, s_] := {(-1105 Cos[r] Sqrt[4514 - 94 Cos[46 t]] Cos[       24 t + \[Theta]] +          2 Sin[r] (1105 Cos[              23 t] (Sqrt[1105] Cos[24 t + \[Theta]] Sin[s] -                23 Sqrt[2] Cos[s] Sin[24 t + \[Theta]]) +             24 Sin[23 t] (1105 Sqrt[2] Cos[s] Cos[24 t + \[Theta]] +                46 Sqrt[1105] Sin[s] Sin[24 t + \[Theta]])))/(-2210 Sqrt[           2257 - 47 Cos[46 t]] + 53040 Sqrt[2] Cos[s] Sin[r] +          1105 Cos[r] Sqrt[4514 - 94 Cos[46 t]] Sin[23 t] -          47 Sqrt[1105] Sin[r] Sin[s] Sin[
       46 t]), -((1105 Sqrt[2] Cos[             s] (-47 Cos[t + \[Theta]] + Cos[47 t + \[Theta]]) Sin[r] +            2208 Sqrt[1105] Cos[24 t + \[Theta]] Sin[r] Sin[s] Sin[23 t] +            1105 (Cos[r] Sqrt[4514 - 94 Cos[46 t]] -               2 Sqrt[1105] Cos[23 t] Sin[r] Sin[s]) Sin[
         24 t + \[Theta]])/(-2210 Sqrt[2257 - 47 Cos[46 t]] +            53040 Sqrt[2] Cos[s] Sin[r] +            1105 Cos[r] Sqrt[4514 - 94 Cos[46 t]] Sin[23 t] -            47 Sqrt[1105] Sin[r] Sin[s] Sin[46 t])), ((Cos[r] Cos[23 t])/          Sqrt[2] + (Sqrt[2257 - 47 Cos[46 t]] Sin[r] Sin[s])/(2 Sqrt[             1105]))/(1 - (Cos[r] Sin[23 t])/          Sqrt[2] + (Sin[             r] (-53040 Sqrt[2] Cos[s] + 
          47 Sqrt[1105] Sin[s] Sin[46 t]))/(2210 Sqrt[             2257 - 47 Cos[46 t]]))}

Two tips,

• not 100% sure but it seems that variation in theta is just a rotation around z. One can generate ParametricPlot3D and later dynamicaly Rotate or change the ViewPoint, should be faster.

Don't have time for math now (cries internally) but random checks seem to support that claim:

    (
      RotationMatrix[.1, {0, 0, 1}].expr[r, t, 0., s] -       expr[r, t, .1, s]
     ) // N //   ReplaceAll[{r -> .3, s -> RandomReal[], t -> RandomReal[]}] // Chop 
    (* {0,0,0}*)

• your figure is symetric and we can use that to focus plotting only on unique part instead of sampling over whole t and s.

Here is what we get:

With[{r = .03, viewpoint = {0, 3, 0}, 
  cols = RGBColor /@ {"#f54123", "#0098d8", "#0b3536"}},

 static = 
  First@ParametricPlot3D[
    expr[r, t, 0, s], {t, 0, 2 \[Pi]/23.}, {s, 0, 2 \[Pi]}
    , PlotPoints -> 50, PlotRange -> 2.7, ViewPoint -> viewpoint, 
    PlotStyle -> White, Mesh -> None, ViewAngle -> \[Pi]/9., 
    ViewVertical -> {0, 0, -1}, Boxed -> False, 
    Background -> cols[[-1]], 
    Lighting -> {{"Point", cols[[1]], {3/4, 0, 0}}, {"Point", 
       cols[[2]], {-3/4, 0, 0}}, {"Ambient", cols[[-1]], viewpoint}}, 
    ImageSize -> 540];

 Animate[
  Graphics3D[
   GeometricTransformation[
    static, 
    Table[RotationTransform[t + Dynamic@\[Theta], {0, 0, 1}], {t, 0, 
      2 Pi - 2 Pi/24, 2 Pi/24}]]

   , PlotRange -> 2.7, ViewPoint -> viewpoint, Axes -> None, 
   ViewAngle -> \[Pi]/9, ViewVertical -> {0, 0, -1}, Boxed -> False, 
   Background -> cols[[-1]],
   ImageSize -> 540]
  ,
  {\[Theta], 0., -Pi/2}
  ]
 ]

If i understand correctly, you use ParametricPlot to create the 'tubes'. Why not use the Tube primitive? (replace the line of a parametric plot with a tube with something like:

plot /. Line[x___] :> Tube[x,0.1]

Should be faster to render?

Actually, it can be done, by specifying a list of 'r' values:

data = Transpose@Table[{{x, 0, 0}, 2 + Cos[x]}, {x, 0, 4 Pi, Pi/20.0}];
Graphics3D[{CapForm[None], Tube @@ data}]

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